What is KDE Connect?

KDE Connect is a project to communicate across all your devices. For example, with KDE Connect you can receive your phone notifications on your desktop computer, control music playing on your phone from your desktop, or use your phone as a remote control for your desktop. To achieve this, KDE Connect:

implements a secure communication protocol over the network, and allows any developer to create plugins on top of it.

Installation

You will most likely find the KDE Connect desktop component as a package in your distribution's repos. If you use a desktop environment other than KDE's Plasma, you might also want to install indicator-kdeconnect, that provides a system tray as a GUI for other desktops.

The app for Android can be found in both the Google Play Store and the free and open store F-Droid.
There was some development of an KDE Connect client app for iOS in 2014 (see source code) and supposedly a client app for Blackberry (where?).

If you use the Firefox browser, this cool extension lets you send links in the browser to the phone app.

If you are a Chrome/Chromium (or compatible) user, this extension lets you "send pages and content from your browser to connected KDE Connect devices, via browser action or context menu." See its Github page for installation instructions.

Troubleshooting

I have two devices running KDE Connnect on the same network, but they can't see each other

KDE Connect uses dynamic ports in the range 1714-1764 for UDP and TCP. So if you are behind a firewall, make sure to open this port range for both TCP and UDP. Otherwise, make sure your network is not blocking UDP broadcast packets.